Derive the wave equation governing the velocity potential $\phi$ for linearized sound waves in a compressible inviscid fluid. How is the pressure disturbance related to the velocity potential?

A semi-infinite straight tube of uniform cross-section is aligned along the positive $x$-axis with its end at $x=-L$. The tube is filled with fluid of density $\rho_{1}$ and sound speed $c_{1}$ in $-L<x<0$ and with fluid of density $\rho_{2}$ and sound speed $c_{2}$ in $x>0$. A piston at the end of the tube performs small oscillations such that its position is $x=-L+\epsilon e^{i \omega t}$, with $\epsilon \ll L$ and $\epsilon \omega \ll c_{1}, c_{2}$. Show that the complex amplitude of the velocity potential in $x>0$ is

$$-\epsilon c_{1}\left(\frac{c_{1}}{c_{2}} \cos \frac{\omega L}{c_{1}}+i \frac{\rho_{2}}{\rho_{1}} \sin \frac{\omega L}{c_{1}}\right)^{-1} .$$

Calculate the time-averaged acoustic energy flux in $x>0$. Comment briefly on the variation of this result with $L$ for the particular case $\rho_{2} \ll \rho_{1}$ and $c_{2}=\mathrm{O}\left(c_{1}\right)$.